For \(|\lambda |<1\)  we also have the formula \(\omega _2(\lambda )=2\int _0^{\pi /2}(1-\lambda \sin ^2\theta )^{-1/2}{\rm d}\theta \)  (see [1]}, Thm. 5.9). For \(|1-\lambda |<1\)  this yields \(\omega _2(1-\lambda )=2\int _0^{\pi /2}(\sin ^2\theta +\lambda \cos ^2\theta )^{-1/2}{\rm d}\theta \) , whence after approximating \(\sin \theta ,\cos \theta \)  by \(\theta ,1\)  near \(\theta =0\) , we easily obtain
\(\omega _2(1-\lambda )=-\log \lambda +O(1)\)  as \(\lambda \in {R}\)  tends to \(0^+\) .
